2.29.33 Problem 93

Internal problem ID [13754]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 93
Date solved : Friday, December 19, 2025 at 11:54:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y&=0 \\ \end{align*}

2.29.33.1 ✓ Maple. Time used: 0.010 (sec). Leaf size: 53

ode:=x*diff(diff(y(x),x),x)+(x^n+1-n)*diff(y(x),x)+b*x^(2*n-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x^{n}}{2 n}} \left (c_1 \sinh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )+c_2 \cosh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )\right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
   -> Trying a Liouvillian solution using Kovacics algorithm 
      A Liouvillian solution exists 
      Group is reducible or imprimitive 
   <- Kovacics algorithm successful 
<- Equivalence, under non-integer power transformations successful
 

2.29.33.2 ✓ Mathematica. Time used: 0.034 (sec). Leaf size: 53

ode=x*D[y[x],{x,2}]+(x^n+1-n)*D[y[x],x]+b*x^(2*n-1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\frac {\left (\sqrt {1-4 b}+1\right ) x^n}{2 n}} \left (c_2 e^{\frac {\sqrt {1-4 b} x^n}{n}}+c_1\right ) \end{align*}

2.29.33.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
b = symbols("b") 
n = symbols("n") 
y = Function("y") 
ode = Eq(b*x**(2*n - 1)*y(x) + x*Derivative(y(x), (x, 2)) + (-n + x**n + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False